Analysis of etching at a solid-solid interface

Abstract

We present a method to derive an analytical expression for the roughness of an eroded surface whose dynamics are ruled by cellular automaton. Starting from the automaton, we obtain the time evolution of the height average and height variance (roughness). We apply this method to the etching model in 1 + 1 dimensions, and then we obtain the roughness exponent. Using this in conjunction with the Galilean invariance we obtain the other exponents, which perfectly match the numerical results obtained from simulations. These exponents are exact, and they are the same as those exhibited by the Kardar-Parisi-Zhang (KPZ) model for this dimension. Therefore, our results provide proof for the conjecture that the etching and KPZ models belong to the same universality class. Moreover, the method is general, and it can be applied to other cellular automata models.

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• The Kardar-Parisi-Zhang exponents for the $2+1$ dimensions
• The fractal geometry of growth: fluctuation-dissipation theorem and hidden symmetry
• Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain
• Universal scaling relation for growth phenomena
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Restoring the fluctuation-dissipation theorem in Kardar-Parisi-Zhang universality class through a new emergent fractal dimension
• A geometrical interpretation of critical exponents
• Fractality in resistive circuits: The Fibonacci resistor networks
• Scaling, Fractal Dynamics and Critical Exponents: Application in a non-integer dimensional Ising model